Symplectic symmetry of quadratic-band-touching Hamiltonians in two dimensions

TL;DR

This paper identifies a symplectic symmetry, USp(2N), in quadratic-band-touching Hamiltonians in two dimensions, analogous to the known Lorentz-invariant Dirac Hamiltonian symmetry, and explores its implications for interactions and lattice symmetries.

Contribution

It reveals the symplectic group USp(2N) as the internal symmetry of quadratic-band-touching Hamiltonians and analyzes the effects of interactions respecting this symmetry.

Findings

The internal symmetry of quadratic-band-touching Hamiltonians is USp(2N).

Interacting theories with USp(2N) symmetry allow two independent interaction terms.

Symmetry can be preserved or spontaneously broken to USp(N) x USp(N) under relevant interactions.

Abstract

The internal low-energy symmetry of the massless Lorentz-invariant Dirac Hamiltonian in $2+1$ dimensions is known to be $O(2N)$, where $N$ is the number of two-component Dirac fermions. Here we point out that there exists an analogous internal symmetry of the single-particle quadratic-band-touching Hamiltonian in two spatial dimensions, and it is the unitary symplectic group, $USp(2N)$. All fermionic bilinears belong to one of the three small irreducible representations of this group. The interacting theory that respects the $USp(2N)$ symmetry and the spatial rotations is constructed and found to allow two independent interaction terms. When these interactions are infrared-relevant the symplectic symmetry either remains preserved or becomes spontaneously broken to $USp(N) \times USp(N)$. The symmetry in the lattices such as honeycomb to infinite order in the dispersion's expansion in powers of local momentum is given by the overlap of the symplectic and the orthogonal groups. We show that this overlap is $O(2N) \bigcap USp(2N) = U(N)$.